That is a valid point but I think that is outside the scope of dimensional analysis. Distinguishing scalars from vectors (or just different rank tensors in gene

That is a valid point but I think that is outside the scope of dimensional analysis. Distinguishing scalars from vectors (or just different rank tensors in general) is usually obvious, because they have a different number of components, and nobody even tries to add them together. If you do try, you will get a result depends on your coordinate system, a big no-no in physics. In some cases, like vectors and axial vectors, it is less obvious because (in 3 spatial dimensions), they actually have the same number of components. But still, if you add them together you get something that does not transform in a sensible way under reflections.